Thanks for sharing the information, it's just amazing (-.-)

I found another issue with the compiled function implementation: i had already assigned a numerical value to "angle" in my notebook and was not aware of it, but when you do a clean restart and this is not the case then the compiled function will not build correctly. In consequence the angle variable has also to be put in the interface to the compiled function. So below is a corrected version and I checked it with a clean restart.

f1 = Compile[{{xy, _Real, 1}}, 
   Module[{cs = Cos[xy[[3]]], sn = Sin[xy[[3]]], x = xy[[1]], 
     y = xy[[2]]}, {N[x cs + (x^2 - y) sn], 
     N[x sn - (x^2 - y) cs], xy[[3]]}]];

kamtorus2[ angle_, step_, ende_, points_] := 
 Module[{ x, xold, y, orbit},
  For[orbit = 0, orbit < ende, orbit += step,
   pts = AppendTo[pts, 
      NestWhileList[f1, {orbit/3., orbit/3., angle}, 
       Total[Abs[#]] < 2. 10^38 &, 1, points ]];
   ]]

pts = {};
Timing[kamtorus2[1.3, 0.05, 1.5, 200]]
ListPlot[Flatten[pts, 1][[All, 1 ;; 2]], AspectRatio -> 1]

Now it should compile and run smoothly and the output reacts to the setting of angle

Here is a more Mathematica style implementation (which runs faster):

kamtorus2[ angle_, step_, ende_, points_] := 
 Module[{ x, xold, y, orbit},
  For[orbit = 0, orbit < ende, orbit += step,
   pts = AppendTo[pts, 
      NestWhileList[f1, orbit/3. {1, 1}, Total[Abs[#]] < 2. 10^38 &, 
       1, points ]];
   ]]

pts = {};
Timing[kamtorus2[1.3, 0.05, 1.5, 200]]
ListPlot[Flatten[pts, 1], AspectRatio -> 1]

Below is a somewhat naive implementation.

kamtorus[ angle_, step_, ende_, points_] := 
 Module[{ x, xold, y, orbit},
  For[orbit = 0, orbit < ende, orbit += step,
   x = orbit/3.;
   y = x;
   ptno = 0;
   While[Abs[x] < 1. 10^38 && Abs[y] < 1. 10^38 &&  ptno < points,
    xold = x;
    x = N[ x        Cos[angle] +     (x^2 - y) Sin[angle]];
    y =  N[ xold Sin[angle] - (xold^2 - y) Cos[angle]];
    pts = AppendTo[pts, {x, y}];
    ptno++
    ];
   ]
  ]

pts = {};
Timing[kamtorus[1.3, 0.05, 1.5, 200]]
ListPlot[pts, AspectRatio -> 1]